Particle accretion and release in flows of suspensions

We consider a pressure-driven, one-dimensional, advective flow of a suspension through a porous domain. Suspended particles are filtered out, changing the porosity. These changes affect the filtration velocity through a porosity-dependent permeability factor. The reaction mechanism includes an accretion term and a term which allows trapped particles to return to the mobile phase. We obtain numerical solutions on a bounded domain and illustrate both contamination and remediation effects, and we consider the existence of traveling wave fronts in an infinite domain. For the latter, accretion rates are characterized that permit such fronts.     

Mathematical and numerical modelling of a circular cross-flow filtration module

We present a computational modelling framework to assess the fluid dynamic behaviour of a circular cross-flow filtration module for water purification. We study two modelling approaches, namely, the Navier-Stokes-Darcy and the one-domain models, that provide a different characterization of the flow in the interfacial region between the feed domain and the membrane surface. Extensive comparison of the numerical results obtained by the two approaches highlights significant differences in the predicted fluid tangential velocity on the membrane surface. Numerical modelling permits to gain a deeper understanding of the flow behaviour than the sole experimental work, e.g., by identifying Dean vortices inside the feed domain and by relating them to geometrical and flow characteristics. This study lays the basis for the optimization of the circular cross-flow filtration module.     

Analysis of stationary filtration problems with a multivalued law in the presence of several point sources

We suggest a generalized statement of stationary filtration problems for an incompressible fluid obeying a multivalued filtration law with limit gradient in an arbitrary bounded nonone-dimensional domain in the presence of several point sources modeled by delta functions. The function determining the filtration law is assumed to grow linearly at infinity. The problems are stated in the form of an integral variational inequality of the second kind. We prove existence theorems and study the properties of solutions. To solve the problem, we suggest an iteration method whose each step essentially amounts to solving the Dirichlet problem for the Poisson equation.     

Fluid-structure interaction in porous media for loaded filter pleats

Currently, the known simulation efforts with respect to fluid-structure interaction (FSI) are mainly restricted to the study of flow interacting with deformable solid bodies. On the other hand, there is extensive literature on simulation of flow through porous media. In particular, algorithms and software for CFD simulations of filtration processes in the case of rigid filtration medium were presented earlier by Fraunhofer ITWM, see, e.g., [1, 2]. In these papers the deformation of the solid skeleton is neglected. In many cases of water filtration, filtration for hydraulic applications, and even in certain air filtration regimes, the fluid pressure can be quite high, and the deformation of the pleats can be an issue. The deflection of pleats and its effect on the filtration process merits examination because under operating conditions (and especially after a partial loading of the media) the pleats often cannot be anymore considered as rigid porous media. Therefore, in this paper, the deflection is considered for clean media, as well as for partially loaded media. A new model describing the coupled flow and deformation, and corresponding numerical results are presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variable boundary method for modelling two dimensional free surface flows with moving boundaries

A coordinate transformation method is proposed for modelling unsteady, depth-averaged shallow water equations for a open channel flow with moving lateral boundaries. The transformation technique which maps the changing domain onto a fixed domain and solves the governing equations in the mapped domain, facilitates the numerical treatment of an irregular boundary configuration. The transformed equations are solved on a staggered grid with a conditionally stable, explicit finite difference scheme. Several numerical experiments are carried out corresponding to different situations, viz., flow with constant discharge, flow with constant discharge and a closed boundary at the downstream, flow in a converging channel with constant discharge and finally flow with varying discharge. The experiments are used to verify the model ability to predict free surface elevation, circulatory pattern and displacement of the boundaries. The simulated results such as displaced area, depth, displacement–time and flow-field are used to evaluate the effects of excess discharge at the upstream on the movement of lateral boundaries.     

A second‐order backward difference time‐stepping scheme for penalized Navier–Stokes equations modeling filtration through porous media

We investigate the stability and convergence of a fully implicit, linearly extrapolated second‐order backward difference time‐stepping scheme for the penalized Navier–Stokes equations modeling filtration through porous media. In the penalization approach, an extended Navier–Stokes equation is used in the entire computational domain with suitable resistance terms to mimic the presence of porous medium. It is widely used as an alternative to the heterogeneous approach in which different types of partial differential equations (PDEs) are used in fluid and porous subregions along with suitable continuity conditions at the interface. However, the introduction of extra resistance terms makes the penalized Navier–Stokes equations more nonlinear. We prove that the linearly extrapolated scheme is unconditionally stable and derive optimal order error estimates without any stability condition. To show feasibility and applicability of the approach, it is used to numerically solve a passive control problem in which flow around a solid body is controlled by adding porous layers on the surface. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 681–705, 2016     

Run up flow of an incompressible micropolar fluid between parallel plates – A state space approach

In this paper, we study the run up flow of an incompressible micropolar fluid between two horizontal infinitely long parallel plates. Initially a flow of the fluid is induced by a constant pressure gradient until steady state is reached. After the steady state is reached, the pressure gradient is suddenly withdrawn while the two plates are impulsively started with different velocities in their own plane. Using the Laplace transform technique and adopting the state space approach, we obtain the velocity and microrotation components in Laplace transform domain. A standard numerical inversion procedure is used to find the velocity and microrotation in space-time domain for various values of time, distance, material parameters and pressure gradient. The variation of velocity and microrotation components is studied and the results are illustrated through graphs. It is observed that the micropolarity parameter has a decreasing effect on velocity component. It is also found that as the gyration parameter increases there is a decrease in microrotation component and an increase in velocity component.     

Asymptotics of the Teichmüller harmonic map flow

The Teichmüller harmonic map flow, introduced by Rupflin and Topping (2012)  [11], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.     

Gas emission into the excavated space during the mining of coal deposits

We consider the problem of nonsteady filtration of a gas in a bounded domain with an unknown moving boundary. We obtain a closed-form solution both for the case when the moving boundary is smaller than the boundary of the filtration region and in the case when the two boundaries coincide. We study the regularities of variation of gas emission, pressure distribution, and advance of the free boundary as functions of time and the filtration properties of the coal vein. Four figures. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 81–85, 1991.     

Numerical solution of three-dimensional problems of filtration consolidation with regard for the influence of technogenic factors by the method of radial basis functions

We use a meshless method to find an approximate solution of the problem that describes a mathematical model of the filtration consolidation process in a three-dimensional domain. It is based on the collocation method using radial basis functions. The performed numerical experiments testify to the efficiency of the proposed approach.     

A gradient flow approach to an evolution problem arising in superconductivity

We study an evolution equation proposed by Chapman, Rubinstein, and Schatzman as a mean‐field model for the evolution of the vortex density in a superconductor. We treat the case of a bounded domain where vortices can exit or enter the domain. We show that the equation can be derived rigorously as the gradient flow of some specific energy for the Riemannian structure induced by the Wasserstein distance on probability measures. This leads us to some existence and uniqueness results and energy‐dissipation identities. We also exhibit some “entropies” that decrease through the flow and allow us to get regularity results (solutions starting in L^p, p > 1, remain in L^p). © 2007 Wiley Periodicals, Inc.     

Four 2-nodal solutions for a semilinear elliptic equation in a finite strip with a hole

In this paper, we study the decomposition of the filtration of the Nehari manifold via the variation of domain shape. We use this result to prove that the semilinear elliptic equation in a finite strip with hole has at least four 2-nodal solutions (solutions with precisely two nodal domains). Furthermore, we can describe the bump location of these solutions.     

On the nonexistence of a global nontrivial subsonic solution in a 3D unbounded angular domain

In this paper, under some assumptions on the flow with a low Mach number, we study the nonexistence of a global nontrivial subsonic solution in an unbounded domain Ω which is one part of a 3D ramp. The flow is assumed to be steady, isentropic and irrotational, namely, the movement of the flow is described by the potential equation. By establishing a fundamental a priori estimate on the solution of a second order linear elliptic equation in Ω with Neumann boundary conditions on Ω and Dirichlet boundary value at some point of Ω, we show that there is no global nontrivial subsonic flow with a low Mach number in such a domain Ω.     

Asymptotic approximation of the solution of Stokes equations in a domain with highly oscillating boundary

We consider a viscous incompressible flow in an infinite horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities which size depends on a small parameter, and with a fixed height. We assume that the flow is governed by the stationary Stokes equations. Using a boundary layer corrector we derive and analyze a first order asymptotic approximation of the flow.      

Analytical solutions for seepage near material boundaries in dam cores: The Davison-Kalinin problems revisited

Steady Darcian seepage through a dam core and adjacent shells is analytically studied. By conformal mappings of the pentagon in the hodograph plane and triangle in the physical plane flow through a low-permeable dam core is analyzed. Mass-balance conjugation of flow in the core and downstream highly-permeable shell of the embankment is carried out by matching the seepage flow rates in the two zones assuming that all water is intercepted by a toe-drain. Seepage refraction is studied for a wedge-shaped domain where pressure and normal components of the Darcian velocities coincide on the interface between the core and shell. Mathematically, the problem of R-linear conjugation (the Riemann-Hilbert problem) is solved in an explicit form. As an illustration, flow to a semi-circular drain (filter) centered at the triple point (contact between the core, shell and impermeable base) is studied. A piece-wise constant hydraulic gradient in two adjacent angles making a two-layered wedge (the dam base at infinity) is examined. Essentially 2-D seepage in a domain bounded by an inlet constant head segment, an outlet seepage-face curve, a horizontal base and with a straight tilted interface between two zones (core and shell) is investigated. The flow net, isobars, and isotachs in the core and shell are reconstructed by computer algebra routines as functions of hydraulic conductivities of two media, the angle of tilt and the hydraulic head value at a specified point.     

Finite element error estimates for 3D exterior incompressible flow with nonzero velocity at infinity

We consider a stationary incompressible Navier–Stokes flow in a 3D exterior domain, with nonzero velocity at infinity. In order to approximate this flow, we use the stabilized P1–P1 finite element method proposed by Rebollo (Numer Math 79:283–319, 1998). Following an approach by Guirguis and Gunzburger (Model Math Anal Numer 21:445–464, 1987), we apply this method to the Navier–Stokes system with Oseen term in a truncated exterior domain, under a pointwise boundary condition on the artificial boundary. This leads to a discrete problem whose solution approximates the exterior flow, as is shown by error estimates.     

具有两种流体的轴对称渗流的一个自由边界问题

本文处理带有两种流体的轴对称的一个自由边界问题，其中在渗流区域的上部都是油，下部是水，这是同时取油注水的一个数学模型。下面，我们将用复分析方法求出此自由边界问题的一个解，并证明其解的唯一性。     

Multiplicity and concentration of solutions for nonhomogeneous elliptic equations in multi-bump domains

In this paper, we study the decomposition of the filtration of the Nehari manifold via the variation of domain shape. Furthermore, we use this result to prove that the nonhomogeneous elliptic equations in an m$m$-bump domain have at least m+1$m+1$ positive solutions.     

Sakiadis流动的一致有效级数解

通过引入一个变换式,克服了Sakiadis流动中半无限大流动区域以及无穷远处渐近边界条件所带来的数学处理上的困难.基于泛函分析中的不动点理论,采用不动点方法求解了变换后的非线性微分方程,获得了Sakiadis流动的近似解析解.该近似解析解用级数的形式来表达并在整个半无限大流动区域内一致有效.     

Identification of a coefficient in the leading term of a pseudoparabolic equation of filtration

The identification problem of a leading coefficient in a linear pseudoparabolic equation is examined under an integral overdetermination condition on the boundary of a domain. We prove a local existence and uniqueness theorem of a strong solution. In a particular case the results obtained allow us to solve a problem of determining the rate of fluid exchange between fissures and pores in a model filtration equation for a fissured medium.